Suppose $\pi :E^{m+k} \to M^m$ if a smooth vector bundle of rank $k$ between to smooth manifolds and $\pi' : E'^{m+k'} \to M^m$ be another one. What I want to understand is how $\overline{\pi}: E \otimes E' \to M$ is defined. I know that as a set $E \otimes E' = \bigcup\limits_{p \in M}E_p \otimes E'_p$. My problem is how can we define the smooth structure over this space. I know that we have to use the a prior topology defining the trivialization $\overline{\varphi}: \overline{\pi}^{-1}(U) \to U \times (\mathbb{R}^k \otimes \mathbb{R}^h)$ using $\varphi:\pi^{-1}(U)\to U \times \mathbb{R}^k$ and $\varphi':\pi'^{-1}(U)\to U \times \mathbb{R}^h$. What I want to prove is that:
- $\overline{\varphi}_i(\overline{\pi}^{-1}(U_i \cap U_j))$ is open
- $\overline{\varphi}_{ij}$ is a smooth function (here $\overline{\varphi}_{ij}=\overline{\varphi}_j \circ \overline{\varphi}_i^{-1}$)
- with the topology induced by this atlas $E \otimes E'$ is Hausdorff and has a countable basis. in such a way we can say $E \otimes E'$ is a smooth manifold.
I appreciate also a reference where these details are filled.